Finite Group Databasebeta

← #29730368

Subgroups — #29730368

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840044C211yesmaximal, characteristic, complemented
2C2×#16839834C211yesmaximal, characteristic
3#21450728C211yesmaximal, characteristic
4#21447256C211yesmaximal, characteristic, complemented
5#21447255C211yesmaximal, characteristic, complemented
6#21447022C211yesmaximal, characteristic, complemented
7#21447015C211yesmaximal, characteristic, complemented
8#21446793C211yesmaximal, characteristic, complemented
9#21446587C211yesmaximal, characteristic, complemented
10#21446250C211yesmaximal, characteristic, complemented
11#21446240C211yesmaximal, characteristic, complemented
12#21445850C211yesmaximal, characteristic, complemented
13#21445842C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2563C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21#16840044—24no—
22#16839834—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35#16839693—21no—
36#16839676—21no—
37#16839650—21no—
38#16839650—21no—
39#16839628C2211yescharacteristic
40#16839622C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#393—21no—
53C61×#393—21no—
54C61×#390—21no—
55C61×#382—21no—
56C61×#380C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194—41no—
65C22×#92418—22no—
66C22×#92418—22no—
67C22×#92418—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519756—22no—
73#14519756—22no—
74#14519756—41no—
75#14519756—41no—
76C4×Dic122—22no—
77C4×Dic122—22no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89#14519710—41no—
90#14519708—22no—
91#14519708—22no—
92#14519708—41no—
93C4⋊#92418D414yes—
94#14519700—22no—
95#14519700—22no—
96#14519700—22no—
97#14519700—22no—
98C22⋊#92418—22no—
99C22⋊#92418—22no—
100#14519688—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61—21no—
106C22×D4×C61D411yescharacteristic
107C23×C4×C61C2311yescharacteristic
108C23×C4×C61C2311yescharacteristic
109C4⋊Q8×C61—22no—
110C4⋊Q8×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61D411yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61C2×D411yescharacteristic
133C24×C61C2×D411yescharacteristic
134C2×Q8×C61—42no—
135C2×Q8×C61—81no—
136C2×Q8×C61—81no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—41no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122#4112yes—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—42no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—81no—
184C22⋊C4×C61—81no—
185C22⋊C4×C61—81no—
186C22⋊C4×C61—81no—
187C42×C61—42no—
188C42×C61C2×D412yes—
189C42×C61#4112yes—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758196—42no—
201#11758196—42no—
202#11758196—42no—
203C4×Dic61—42no—
204#11758194—44no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—44no—
215C2×#92418C2×D412yes—
216C2×#92418C2×D412yes—
217C2×#92418#4112yes—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61#13712yes—
232C23×C61C22×D411yescharacteristic
233Q8×C61—84no—
234D4×C61—44no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61#12112yes—
266C2×C4×C61#12112yes—
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×C4×C612−1+411yescharacteristic
273C2×Dic61—22no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289#92418—22no—
290#92418—22no—
291#92418—22no—
292#92418—41no—
293#25356—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61C2×#12111yescharacteristic
309C22×C61#54611yescharacteristic
310C22×C61#54711yescharacteristic
311C22×C61C2×#13711yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61#52911yescharacteristic
324C4×C61#54211yescharacteristic
325C4×C61#54611yescharacteristic
326C4×C61#54811yescharacteristic
327Dic61—42no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#475—611no—
332C2×#435—611no—
333#2563D6111yescharacteristic
334#2536—611no—
335#2077—611no—
336#2077—611no—
337#2036—611no—
338#2035—611no—
339#1993—611no—
340#1949—611no—
341#1894—611no—
342#1892—611no—
343#1809—611no—
344#1807—611no—
345#1763—611no—
346C2×C61—42no—
347C2×C61—81no—
348C2×C61#242711yescharacteristic
349C2×C61#305611yescharacteristic
350C2×C61#305711yescharacteristic
351C2×C61#331611yescharacteristic
352C2×C61#332111yescharacteristic
353C2×C61#334411yescharacteristic
354C2×C61#334811yescharacteristic
355C22×M16—611no—
356C23×C8—611no—
357C2×C4⋊Q8C2×D6111yescharacteristic
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—611no—
360C2×C4×Q8—611no—
361C2×C4×D4C2×D6111yescharacteristic
362#475—1224no—
363#435—1224no—
364C2×#107—611no—
365C2×#106—611no—
366C2×#106—611no—
367C2×#106—611no—
368C2×C4⋊C8—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×C22⋊C8—611no—
374C2×C22⋊C8—611no—
375C2×#97—611no—
376#398—1221no—
377#394—1221no—
378#393—21no—
379#393—21no—
380#390—21no—
381#388—1221no—
382#388—1221no—
383#382—21no—
384#382—611no—
385#380—611no—
386#380—611no—
387#380C2×D6111yescharacteristic
388#380C2×D6111yescharacteristic
389#380C2×D6111yescharacteristic
390#380C2×D6111yescharacteristic
391C4×C4⋊C4C2×D6111yescharacteristic
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400C61#2535611yescharacteristic, complemented, Sylow
401C22×Q8—21no—
402C22×Q8—611no—
403C22×D4C22⋊D6111yescharacteristic
404C23×C4C22×D6111yescharacteristic
405C23×C4C22×D6111yescharacteristic
406C2×M16—1222no—
407C2×M16—1222no—
408C2×M16—2441no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—611no—
413C22×C8—1222no—
414C22×C8—1222no—
415C22×C8—2441no—
416C4⋊Q8—22no—
417C4⋊Q8—22no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C22⋊Q8—2441no—
421C22⋊Q8—2441no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×D4—22no—
425C4×D4—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4C22⋊D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4C22⋊D6111yescharacteristic
447C2×C22⋊C4C22⋊D6111yescharacteristic
448C2×C42—21no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451#107—2441no—
452#106—1222no—
453#106—1222no—
454#106—2441no—
455C4⋊C8—614no—
456#103—1222no—
457#103—1222no—
458#103—1222no—
459#103—1222no—
460C22⋊C8—1222no—
461C22⋊C8—1222no—
462#97—1222no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24C2×C22⋊D6111yescharacteristic
473C24C2×C22⋊D6111yescharacteristic
474C2×Q8—42no—
475C2×Q8—81no—
476C2×Q8—81no—
477C2×Q8—612no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—41no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511M16—2444no—
512C2×C8—612no—
513C2×C8—612no—
514C2×C8—612no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—2444no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—42no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—2442no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—81no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C42—42no—
555C42—2442no—
556C42D4×D6112yes—
557C42#1175822412yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23#1451980812yes—
572C23C22×C22⋊D6111yescharacteristic
573Q8—84no—
574Q8—1222no—
575Q8—1222no—
576Q8—2441no—
577Q8—2441no—
578D4—44no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—1222no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451974912yes—
622C2×C4#1451974912yes—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4#41×D6111yescharacteristic
626C2×C4C2×#1175822411yescharacteristic
627C2×C4#1451982311yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—1222no—
630C8—1222no—
631C8—1222no—
632C8—2441no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22C2×#1451974911yescharacteristic
648C22#1684040011yescharacteristic
649C22#1684041111yescharacteristic
650C22C2×#1451980811yescharacteristic
651C22#1684056011yescharacteristic
652C22#1684057311yescharacteristic
653C22#1684057311yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—2442no—
666C4#1684029111yescharacteristic
667C4#1684038111yescharacteristic
668C4#1684040511yescharacteristic
669C4#1684042011yescharacteristic
670C2—42no—
671C2—81no—
672C2#2144939911yescharacteristic
673C2#2145544511yescharacteristic
674C2#2145545511yescharacteristic
675C2#2145792411yescharacteristic
676C2#2145799911yescharacteristic
677C2#2145826811yescharacteristic
678C2#2145833411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2563C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2563D61
hypocenterC61#25356
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2563

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840044C2semidirect
4. #21447256C2semidirect
5. #21447255C2semidirect
6. #21447022C2semidirect
7. #21447015C2semidirect
8. #21446793C2semidirect
9. #21446587C2semidirect
10. #21446250C2semidirect
11. #21446240C2semidirect
12. #21445850C2semidirect
13. #21445842C2semidirect
14. #21445652C2semidirect
400. C61#25356semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400442
C2×#168398342
#214507282
#214472562
#214472552
#214470222
#214470152
#214467932
#214465872
#214462502
#214462402
#214458502
#214458422
#214456522
C61×#25632
#2535661